On the spectral problems of Kolmogorov and Rokhlin in the class of mixing automorphisms

TL;DR

This paper investigates the spectral properties of certain mixing automorphisms, showing that specific rank-one constructions lack the group property in their spectrum and that their tensor products have homogeneous spectrum of multiplicity two.

Contribution

It provides new results on the spectral behavior of staircase rank-one automorphisms with particular parameters, highlighting their non-group spectrum and multiplicity properties.

Findings

Spectrum of the automorphism lacks the group property.

Product automorphism has homogeneous spectrum of multiplicity 2.

Results apply to staircase rank-one constructions with parameters growing as j^d, 0<d<0.2.

Abstract

Let $T$ be a staircase rank-one construction with parameters $r_j \sim j^d$, $0<d<0.2$, then its spectrum does not have the group property, and the product $T\otimes T$ has homogeneous spectrum of multiplicity 2.